10 TJ. S. COAST AND GEODETIC SURVEY. ^ 



points of this arc intersect in K\ If we denote this radius 

 by Pn, we have 



X a cos yy a 



^^ cos ip cos (p (1 — e^ sinV)^ ^ 



If the element of length of the meridian is denoted by dm, 

 we obtain 



a{l — e^) dip 



dm= 



(l-e^smVy 



This is an elliptic integral that it is not necessary to 

 evaluate in this place, since we shaU have occasion to 

 employ it only in the differential form. 



DEVELOPMENT OF GENERAL FORMULAS FOR THE POLY- 

 CONIC PROJECTIONS. 



Tissot defines a polyconic projection as one in which 

 the parallels of latitude are represented by arcs of a non- 

 concentric system of circles, with the centers of these 

 various circles lying upon a straight line. This line of 

 centers is generally called the central meridian; but it is 

 not necessarily the central meridian of any given map 

 and in cases does not appear upon the map at all. 



In the following discussion the latitude wiU be denoted 

 by (p, and the longitude out from the central meridian 

 mil be denoted by X. 



In figure 2 let Q If be the arc of a circle that represents 

 a given X on the parallel of latitude <p, with radius 8Q 

 and center at S. Let RM' be an arc of equal X on the 

 parallel of latitude <p + d(p, with radius S^R and center at S\ 

 is the point of intersection of the central meridian and 

 the Equator. Let OS be denoted by s. Then since s is a 

 decreasing function of (p, SS' is equal to —ds. If the 

 angle QSM is denoted by 6, we have 



SP=-ds cos e. 



S'P=-ds sin e. 



M'N=S'M'X Z.M'S'N, 

 But 



Z M'S'N== lOS'M'- lOS'N 



= AOS'M'- Z.OSN- IS'NS, 



